Optimal. Leaf size=25 \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {c}{(a+b x)^3}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {c}{x^3}}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {2 (a+b x)}{5 b \sqrt {\frac {c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 79, normalized size = 3.16 \[ \frac {2 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{5 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 144, normalized size = 5.76 \[ \frac {2 \, {\left (15 \, \sqrt {b c x + a c} a^{2} - \frac {10 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a}{c} + \frac {15 \, \sqrt {b c x + a c} a^{2} c^{2} - 10 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a c + 3 \, {\left (b c x + a c\right )}^{\frac {5}{2}}}{c^{2}}\right )}}{15 \, b c \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.88 \[ \frac {\frac {2 b x}{5}+\frac {2 a}{5}}{\sqrt {\frac {c}{\left (b x +a \right )^{3}}}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 27, normalized size = 1.08 \[ \frac {2 \, {\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{5 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 26, normalized size = 1.04 \[ \frac {2\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,{\left (a+b\,x\right )}^4}{5\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {c}{\left (a + b x\right )^{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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